-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathplot_kmeans_assumptions.html
469 lines (415 loc) · 52.8 KB
/
plot_kmeans_assumptions.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
<!DOCTYPE html>
<!-- data-theme below is forced to be "light" but should be changed if we use pydata-theme-sphinx in the future -->
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" data-content_root="../../" data-theme="light"> <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" data-content_root="../../" data-theme="light"> <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta property="og:title" content="Demonstration of k-means assumptions" />
<meta property="og:type" content="website" />
<meta property="og:url" content="https://scikit-learn/stable/auto_examples/cluster/plot_kmeans_assumptions.html" />
<meta property="og:site_name" content="scikit-learn" />
<meta property="og:description" content="This example is meant to illustrate situations where k-means produces unintuitive and possibly undesirable clusters. Data generation: The function make_blobs generates isotropic (spherical) gaussia..." />
<meta property="og:image" content="https://scikit-learn.org/stable/_static/scikit-learn-logo-small.png" />
<meta property="og:image:alt" content="scikit-learn" />
<meta name="description" content="This example is meant to illustrate situations where k-means produces unintuitive and possibly undesirable clusters. Data generation: The function make_blobs generates isotropic (spherical) gaussia..." />
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Demonstration of k-means assumptions — scikit-learn 1.4.2 documentation</title>
<link rel="canonical" href="https://scikit-learn.org/stable/auto_examples/cluster/plot_kmeans_assumptions.html" />
<link rel="shortcut icon" href="../../_static/favicon.ico"/>
<link rel="stylesheet" href="../../_static/css/vendor/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="../../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="../../_static/copybutton.css" type="text/css" />
<link rel="stylesheet" href="../../_static/plot_directive.css" type="text/css" />
<link rel="stylesheet" href="../../https://fonts.googleapis.com/css?family=Vibur" type="text/css" />
<link rel="stylesheet" href="../../_static/jupyterlite_sphinx.css" type="text/css" />
<link rel="stylesheet" href="../../_static/sg_gallery.css" type="text/css" />
<link rel="stylesheet" href="../../_static/sg_gallery-binder.css" type="text/css" />
<link rel="stylesheet" href="../../_static/sg_gallery-dataframe.css" type="text/css" />
<link rel="stylesheet" href="../../_static/sg_gallery-rendered-html.css" type="text/css" />
<link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
<script id="documentation_options" data-url_root="../../" src="../../_static/documentation_options.js"></script>
<script src="../../_static/js/vendor/jquery-3.6.3.slim.min.js"></script>
<script src="../../_static/js/details-permalink.js"></script>
</head>
<body>
<nav id="navbar" class="sk-docs-navbar navbar navbar-expand-md navbar-light bg-light py-0">
<div class="container-fluid sk-docs-container px-0">
<a class="navbar-brand py-0" href="../../index.html">
<img
class="sk-brand-img"
src="../../_static/scikit-learn-logo-small.png"
alt="logo"/>
</a>
<button
id="sk-navbar-toggler"
class="navbar-toggler"
type="button"
data-toggle="collapse"
data-target="#navbarSupportedContent"
aria-controls="navbarSupportedContent"
aria-expanded="false"
aria-label="Toggle navigation"
>
<span class="navbar-toggler-icon"></span>
</button>
<div class="sk-navbar-collapse collapse navbar-collapse" id="navbarSupportedContent">
<ul class="navbar-nav mr-auto">
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../../install.html">Install</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../../user_guide.html">User Guide</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../../modules/classes.html">API</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../index.html">Examples</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" target="_blank" rel="noopener noreferrer" href="https://blog.scikit-learn.org/">Community</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../getting_started.html" >Getting Started</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../tutorial/index.html" >Tutorial</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../whats_new/v1.4.html" >What's new</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../glossary.html" >Glossary</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/developers/index.html" target="_blank" rel="noopener noreferrer">Development</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../faq.html" >FAQ</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../support.html" >Support</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../related_projects.html" >Related packages</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../roadmap.html" >Roadmap</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../governance.html" >Governance</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../about.html" >About us</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</li>
<li class="nav-item dropdown nav-more-item-dropdown">
<a class="sk-nav-link nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">More</a>
<div class="dropdown-menu" aria-labelledby="navbarDropdown">
<a class="sk-nav-dropdown-item dropdown-item" href="../../getting_started.html" >Getting Started</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../tutorial/index.html" >Tutorial</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../whats_new/v1.4.html" >What's new</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../glossary.html" >Glossary</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/developers/index.html" target="_blank" rel="noopener noreferrer">Development</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../faq.html" >FAQ</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../support.html" >Support</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../related_projects.html" >Related packages</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../roadmap.html" >Roadmap</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../governance.html" >Governance</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../about.html" >About us</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</div>
</li>
</ul>
<div id="searchbox" role="search">
<div class="searchformwrapper">
<form class="search" action="../../search.html" method="get">
<input class="sk-search-text-input" type="text" name="q" aria-labelledby="searchlabel" />
<input class="sk-search-text-btn" type="submit" value="Go" />
</form>
</div>
</div>
</div>
</div>
</nav>
<div class="d-flex" id="sk-doc-wrapper">
<input type="checkbox" name="sk-toggle-checkbox" id="sk-toggle-checkbox">
<label id="sk-sidemenu-toggle" class="sk-btn-toggle-toc btn sk-btn-primary" for="sk-toggle-checkbox">Toggle Menu</label>
<div id="sk-sidebar-wrapper" class="border-right">
<div class="sk-sidebar-toc-wrapper">
<div class="btn-group w-100 mb-2" role="group" aria-label="rellinks">
<a href="plot_affinity_propagation.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Demo of affinity propagation clustering algorithm">Prev</a><a href="index.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Clustering">Up</a>
<a href="plot_kmeans_stability_low_dim_dense.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Empirical evaluation of the impact of k-means initialization">Next</a>
</div>
<div class="alert alert-danger p-1 mb-2" role="alert">
<p class="text-center mb-0">
<strong>scikit-learn 1.4.2</strong><br/>
<a href="https://scikit-learn.org/dev/versions.html">Other versions</a>
</p>
</div>
<div class="alert alert-warning p-1 mb-2" role="alert">
<p class="text-center mb-0">
Please <a class="font-weight-bold" href="../../about.html#citing-scikit-learn"><string>cite us</string></a> if you use the software.
</p>
</div>
<div class="sk-sidebar-toc">
<ul>
<li><a class="reference internal" href="#">Demonstration of k-means assumptions</a><ul>
<li><a class="reference internal" href="#data-generation">Data generation</a></li>
<li><a class="reference internal" href="#fit-models-and-plot-results">Fit models and plot results</a></li>
<li><a class="reference internal" href="#possible-solutions">Possible solutions</a></li>
<li><a class="reference internal" href="#final-remarks">Final remarks</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div id="sk-page-content-wrapper">
<div class="sk-page-content container-fluid body px-md-3" role="main">
<div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p><a class="reference internal" href="#sphx-glr-download-auto-examples-cluster-plot-kmeans-assumptions-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code or to run this example in your browser via JupyterLite or Binder</p>
</div>
<section class="sphx-glr-example-title" id="demonstration-of-k-means-assumptions">
<span id="sphx-glr-auto-examples-cluster-plot-kmeans-assumptions-py"></span><h1>Demonstration of k-means assumptions<a class="headerlink" href="#demonstration-of-k-means-assumptions" title="Link to this heading">¶</a></h1>
<p>This example is meant to illustrate situations where k-means produces
unintuitive and possibly undesirable clusters.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Phil Roth <[email protected]></span>
<span class="c1"># Arturo Amor <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
</pre></div>
</div>
<section id="data-generation">
<h2>Data generation<a class="headerlink" href="#data-generation" title="Link to this heading">¶</a></h2>
<p>The function <a class="reference internal" href="../../modules/generated/sklearn.datasets.make_blobs.html#sklearn.datasets.make_blobs" title="sklearn.datasets.make_blobs"><code class="xref py py-func docutils literal notranslate"><span class="pre">make_blobs</span></code></a> generates isotropic
(spherical) gaussian blobs. To obtain anisotropic (elliptical) gaussian blobs
one has to define a linear <code class="docutils literal notranslate"><span class="pre">transformation</span></code>.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.datasets.make_blobs.html#sklearn.datasets.make_blobs" title="sklearn.datasets.make_blobs" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_blobs</span></a>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">1500</span>
<span class="n">random_state</span> <span class="o">=</span> <span class="mi">170</span>
<span class="n">transformation</span> <span class="o">=</span> <span class="p">[[</span><span class="mf">0.60834549</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.63667341</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.40887718</span><span class="p">,</span> <span class="mf">0.85253229</span><span class="p">]]</span>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.make_blobs.html#sklearn.datasets.make_blobs" title="sklearn.datasets.make_blobs" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_blobs</span></a><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">)</span>
<span class="n">X_aniso</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="numpy.dot" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">transformation</span><span class="p">)</span> <span class="c1"># Anisotropic blobs</span>
<span class="n">X_varied</span><span class="p">,</span> <span class="n">y_varied</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.make_blobs.html#sklearn.datasets.make_blobs" title="sklearn.datasets.make_blobs" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_blobs</span></a><span class="p">(</span>
<span class="n">n_samples</span><span class="o">=</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">cluster_std</span><span class="o">=</span><span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">],</span> <span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span>
<span class="p">)</span> <span class="c1"># Unequal variance</span>
<span class="n">X_filtered</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.vstack.html#numpy.vstack" title="numpy.vstack" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">vstack</span></a><span class="p">(</span>
<span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">0</span><span class="p">][:</span><span class="mi">500</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">1</span><span class="p">][:</span><span class="mi">100</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">y</span> <span class="o">==</span> <span class="mi">2</span><span class="p">][:</span><span class="mi">10</span><span class="p">])</span>
<span class="p">)</span> <span class="c1"># Unevenly sized blobs</span>
<span class="n">y_filtered</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">500</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="mi">100</span> <span class="o">+</span> <span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="mi">10</span>
</pre></div>
</div>
<p>We can visualize the resulting data:</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Mixture of Gaussian Blobs"</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_aniso</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_aniso</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Anisotropically Distributed Blobs"</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_varied</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_varied</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_varied</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Unequal Variance"</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_filtered</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_filtered</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_filtered</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Unevenly Sized Blobs"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.suptitle.html#matplotlib.pyplot.suptitle" title="matplotlib.pyplot.suptitle" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span></a><span class="p">(</span><span class="s2">"Ground truth clusters"</span><span class="p">)</span><span class="o">.</span><span class="n">set_y</span><span class="p">(</span><span class="mf">0.95</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_kmeans_assumptions_001.png" srcset="../../_images/sphx_glr_plot_kmeans_assumptions_001.png" alt="Ground truth clusters, Mixture of Gaussian Blobs, Anisotropically Distributed Blobs, Unequal Variance, Unevenly Sized Blobs" class = "sphx-glr-single-img"/></section>
<section id="fit-models-and-plot-results">
<h2>Fit models and plot results<a class="headerlink" href="#fit-models-and-plot-results" title="Link to this heading">¶</a></h2>
<p>The previously generated data is now used to show how
<a class="reference internal" href="../../modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> behaves in the following scenarios:</p>
<ul class="simple">
<li><p>Non-optimal number of clusters: in a real setting there is no uniquely
defined <strong>true</strong> number of clusters. An appropriate number of clusters has
to be decided from data-based criteria and knowledge of the intended goal.</p></li>
<li><p>Anisotropically distributed blobs: k-means consists of minimizing sample’s
euclidean distances to the centroid of the cluster they are assigned to. As
a consequence, k-means is more appropriate for clusters that are isotropic
and normally distributed (i.e. spherical gaussians).</p></li>
<li><p>Unequal variance: k-means is equivalent to taking the maximum likelihood
estimator for a “mixture” of k gaussian distributions with the same
variances but with possibly different means.</p></li>
<li><p>Unevenly sized blobs: there is no theoretical result about k-means that
states that it requires similar cluster sizes to perform well, yet
minimizing euclidean distances does mean that the more sparse and
high-dimensional the problem is, the higher is the need to run the algorithm
with different centroid seeds to ensure a global minimal inertia.</p></li>
</ul>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KMeans</span></a>
<span class="n">common_params</span> <span class="o">=</span> <span class="p">{</span>
<span class="s2">"n_init"</span><span class="p">:</span> <span class="s2">"auto"</span><span class="p">,</span>
<span class="s2">"random_state"</span><span class="p">:</span> <span class="n">random_state</span><span class="p">,</span>
<span class="p">}</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="n">y_pred</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KMeans</span></a><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">common_params</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_pred</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Non-optimal Number of Clusters"</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KMeans</span></a><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="o">**</span><span class="n">common_params</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">X_aniso</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_aniso</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_aniso</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_pred</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Anisotropically Distributed Blobs"</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KMeans</span></a><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="o">**</span><span class="n">common_params</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">X_varied</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_varied</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_varied</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_pred</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Unequal Variance"</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KMeans</span></a><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="o">**</span><span class="n">common_params</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">X_filtered</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_filtered</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_filtered</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_pred</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Unevenly Sized Blobs"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.suptitle.html#matplotlib.pyplot.suptitle" title="matplotlib.pyplot.suptitle" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span></a><span class="p">(</span><span class="s2">"Unexpected KMeans clusters"</span><span class="p">)</span><span class="o">.</span><span class="n">set_y</span><span class="p">(</span><span class="mf">0.95</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_kmeans_assumptions_002.png" srcset="../../_images/sphx_glr_plot_kmeans_assumptions_002.png" alt="Unexpected KMeans clusters, Non-optimal Number of Clusters, Anisotropically Distributed Blobs, Unequal Variance, Unevenly Sized Blobs" class = "sphx-glr-single-img"/></section>
<section id="possible-solutions">
<h2>Possible solutions<a class="headerlink" href="#possible-solutions" title="Link to this heading">¶</a></h2>
<p>For an example on how to find a correct number of blobs, see
<a class="reference internal" href="plot_kmeans_silhouette_analysis.html#sphx-glr-auto-examples-cluster-plot-kmeans-silhouette-analysis-py"><span class="std std-ref">Selecting the number of clusters with silhouette analysis on KMeans clustering</span></a>.
In this case it suffices to set <code class="docutils literal notranslate"><span class="pre">n_clusters=3</span></code>.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">y_pred</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KMeans</span></a><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="o">**</span><span class="n">common_params</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_pred</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Optimal Number of Clusters"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_kmeans_assumptions_003.png" srcset="../../_images/sphx_glr_plot_kmeans_assumptions_003.png" alt="Optimal Number of Clusters" class = "sphx-glr-single-img"/><p>To deal with unevenly sized blobs one can increase the number of random
initializations. In this case we set <code class="docutils literal notranslate"><span class="pre">n_init=10</span></code> to avoid finding a
sub-optimal local minimum. For more details see <a class="reference internal" href="../text/plot_document_clustering.html#kmeans-sparse-high-dim"><span class="std std-ref">Clustering sparse data with k-means</span></a>.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">y_pred</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KMeans</span></a><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">n_init</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span>
<span class="n">X_filtered</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">X_filtered</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_filtered</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_pred</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Unevenly Sized Blobs </span><span class="se">\n</span><span class="s2">with several initializations"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_kmeans_assumptions_004.png" srcset="../../_images/sphx_glr_plot_kmeans_assumptions_004.png" alt="Unevenly Sized Blobs with several initializations" class = "sphx-glr-single-img"/><p>As anisotropic and unequal variances are real limitations of the k-means
algorithm, here we propose instead the use of
<a class="reference internal" href="../../modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianMixture</span></code></a>, which also assumes gaussian
clusters but does not impose any constraints on their variances. Notice that
one still has to find the correct number of blobs (see
<a class="reference internal" href="../mixture/plot_gmm_selection.html#sphx-glr-auto-examples-mixture-plot-gmm-selection-py"><span class="std std-ref">Gaussian Mixture Model Selection</span></a>).</p>
<p>For an example on how other clustering methods deal with anisotropic or
unequal variance blobs, see the example
<a class="reference internal" href="plot_cluster_comparison.html#sphx-glr-auto-examples-cluster-plot-cluster-comparison-py"><span class="std std-ref">Comparing different clustering algorithms on toy datasets</span></a>.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.mixture</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture" class="sphx-glr-backref-module-sklearn-mixture sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GaussianMixture</span></a>
<span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">)</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="n">y_pred</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture" class="sphx-glr-backref-module-sklearn-mixture sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GaussianMixture</span></a><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">X_aniso</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_aniso</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_aniso</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_pred</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Anisotropically Distributed Blobs"</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture" class="sphx-glr-backref-module-sklearn-mixture sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GaussianMixture</span></a><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">X_varied</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_varied</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_varied</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y_pred</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Unequal Variance"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.suptitle.html#matplotlib.pyplot.suptitle" title="matplotlib.pyplot.suptitle" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span></a><span class="p">(</span><span class="s2">"Gaussian mixture clusters"</span><span class="p">)</span><span class="o">.</span><span class="n">set_y</span><span class="p">(</span><span class="mf">0.95</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_kmeans_assumptions_005.png" srcset="../../_images/sphx_glr_plot_kmeans_assumptions_005.png" alt="Gaussian mixture clusters, Anisotropically Distributed Blobs, Unequal Variance" class = "sphx-glr-single-img"/></section>
<section id="final-remarks">
<h2>Final remarks<a class="headerlink" href="#final-remarks" title="Link to this heading">¶</a></h2>
<p>In high-dimensional spaces, Euclidean distances tend to become inflated
(not shown in this example). Running a dimensionality reduction algorithm
prior to k-means clustering can alleviate this problem and speed up the
computations (see the example
<a class="reference internal" href="../text/plot_document_clustering.html#sphx-glr-auto-examples-text-plot-document-clustering-py"><span class="std std-ref">Clustering text documents using k-means</span></a>).</p>
<p>In the case where clusters are known to be isotropic, have similar variance
and are not too sparse, the k-means algorithm is quite effective and is one of
the fastest clustering algorithms available. This advantage is lost if one has
to restart it several times to avoid convergence to a local minimum.</p>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 1.087 seconds)</p>
<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-cluster-plot-kmeans-assumptions-py">
<div class="binder-badge docutils container">
<a class="reference external image-reference" href="https://mybinder.org/v2/gh/scikit-learn/scikit-learn/1.4.X?urlpath=lab/tree/notebooks/auto_examples/cluster/plot_kmeans_assumptions.ipynb"><img alt="Launch binder" src="../../_images/binder_badge_logo4.svg" width="150px" /></a>
</div>
<div class="lite-badge docutils container">
<a class="reference external image-reference" href="../../lite/lab/?path=auto_examples/cluster/plot_kmeans_assumptions.ipynb"><img alt="Launch JupyterLite" src="../../_images/jupyterlite_badge_logo4.svg" width="150px" /></a>
</div>
<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/b05e6cdf6d51481f37bf29b0bb92995e/plot_kmeans_assumptions.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_kmeans_assumptions.ipynb</span></code></a></p>
</div>
<div class="sphx-glr-download sphx-glr-download-python docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/5d2d581a4569eb0718dbdb8abf7cbbdf/plot_kmeans_assumptions.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_kmeans_assumptions.py</span></code></a></p>
</div>
</div>
<p class="rubric">Related examples</p>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example shows differences between Regular K-Means algorithm and Bisecting K-Means."><img alt="" src="../../_images/sphx_glr_plot_bisect_kmeans_thumb.png" />
<p><a class="reference internal" href="plot_bisect_kmeans.html#sphx-glr-auto-examples-cluster-plot-bisect-kmeans-py"><span class="std std-ref">Bisecting K-Means and Regular K-Means Performance Comparison</span></a></p>
<div class="sphx-glr-thumbnail-title">Bisecting K-Means and Regular K-Means Performance Comparison</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example shows characteristics of different linkage methods for hierarchical clustering on ..."><img alt="" src="../../_images/sphx_glr_plot_linkage_comparison_thumb.png" />
<p><a class="reference internal" href="plot_linkage_comparison.html#sphx-glr-auto-examples-cluster-plot-linkage-comparison-py"><span class="std std-ref">Comparing different hierarchical linkage methods on toy datasets</span></a></p>
<div class="sphx-glr-thumbnail-title">Comparing different hierarchical linkage methods on toy datasets</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example shows how to use cross_val_predict together with PredictionErrorDisplay to visuali..."><img alt="" src="../../_images/sphx_glr_plot_cv_predict_thumb.png" />
<p><a class="reference internal" href="../model_selection/plot_cv_predict.html#sphx-glr-auto-examples-model-selection-plot-cv-predict-py"><span class="std std-ref">Plotting Cross-Validated Predictions</span></a></p>
<div class="sphx-glr-thumbnail-title">Plotting Cross-Validated Predictions</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the prior and posterior of a GaussianProcessRegressor with different k..."><img alt="" src="../../_images/sphx_glr_plot_gpr_prior_posterior_thumb.png" />
<p><a class="reference internal" href="../gaussian_process/plot_gpr_prior_posterior.html#sphx-glr-auto-examples-gaussian-process-plot-gpr-prior-posterior-py"><span class="std std-ref">Illustration of prior and posterior Gaussian process for different kernels</span></a></p>
<div class="sphx-glr-thumbnail-title">Illustration of prior and posterior Gaussian process for different kernels</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="In this example we compare the various initialization strategies for K-means in terms of runtim..."><img alt="" src="../../_images/sphx_glr_plot_kmeans_digits_thumb.png" />
<p><a class="reference internal" href="plot_kmeans_digits.html#sphx-glr-auto-examples-cluster-plot-kmeans-digits-py"><span class="std std-ref">A demo of K-Means clustering on the handwritten digits data</span></a></p>
<div class="sphx-glr-thumbnail-title">A demo of K-Means clustering on the handwritten digits data</div>
</div></div><p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
</section>
</section>
</div>
<div class="container">
<footer class="sk-content-footer">
© 2007 - 2024, scikit-learn developers (BSD License).
<a href="../../_sources/auto_examples/cluster/plot_kmeans_assumptions.rst.txt" rel="nofollow">Show this page source</a>
</footer>
</div>
</div>
</div>
<script src="../../_static/js/vendor/bootstrap.min.js"></script>
<script>
window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
ga('create', 'UA-22606712-2', 'auto');
ga('set', 'anonymizeIp', true);
ga('send', 'pageview');
</script>
<script async src='https://www.google-analytics.com/analytics.js'></script>
<script defer data-domain="scikit-learn.org" src="https://views.scientific-python.org/js/script.js">
</script>
<script src="../../_static/clipboard.min.js"></script>
<script src="../../_static/copybutton.js"></script>
<script>
$(document).ready(function() {
/* Add a [>>>] button on the top-right corner of code samples to hide
* the >>> and ... prompts and the output and thus make the code
* copyable. */
var div = $('.highlight-python .highlight,' +
'.highlight-python3 .highlight,' +
'.highlight-pycon .highlight,' +
'.highlight-default .highlight')
var pre = div.find('pre');
// get the styles from the current theme
pre.parent().parent().css('position', 'relative');
// create and add the button to all the code blocks that contain >>>
div.each(function(index) {
var jthis = $(this);
// tracebacks (.gt) contain bare text elements that need to be
// wrapped in a span to work with .nextUntil() (see later)
jthis.find('pre:has(.gt)').contents().filter(function() {
return ((this.nodeType == 3) && (this.data.trim().length > 0));
}).wrap('<span>');
});
/*** Add permalink buttons next to glossary terms ***/
$('dl.glossary > dt[id]').append(function() {
return ('<a class="headerlink" href="#' +
this.getAttribute('id') +
'" title="Permalink to this term">¶</a>');
});
});
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<script src="https://scikit-learn.org/versionwarning.js"></script>
</body>
</html>